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Polynomials with at least one integer root

Problem 269

Published on Saturday, 19th December 2009, 09:00 am; Solved by 426

A root or zero of a polynomial P(x) is a solution to the equation P(x) = 0.
Define Pn as the polynomial whose coefficients are the digits of n.
For example, P5703(x) = 5x3 + 7x2 + 3.

We can see that:

  • Pn(0) is the last digit of n,
  • Pn(1) is the sum of the digits of n,
  • Pn(10) is n itself.

Define Z(k) as the number of positive integers, n, not exceeding k for which the polynomial Pn has at least one integer root.

It can be verified that Z(100 000) is 14696.

What is Z(1016)?